Probability Density Function of the Reverse Gumbel Distribution

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Description

This function computes the probability density of the Reverse Gumbel distribution given parameters (ξ and α) computed by parrevgum. The probability density function is

f(x) = α^{-1} \exp(Y) [\exp(\exp[-\exp(Y)])] \mbox{,}

where

Y = \frac{x - ξ}{α} \mbox{,}

where f(x) is the probability density for quantile x, ξ is a location parameter, and α is a scale parameter.

Usage

1
pdfrevgum(x, para)

Arguments

x

A real value vector.

para

The parameters from parrevgum or vec2par.

Value

Probability density (f) for x.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105–124.

Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 29, CRC Press, Boca Raton, Fla., pp. 546–560.

See Also

cdfrevgum, quarevgum, lmomrevgum, parrevgum

Examples

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# See p. 553 of Hosking (1995)
# Data listed in Hosking (1995, table 29.3, p. 553)
D <- c(-2.982, -2.849, -2.546, -2.350, -1.983, -1.492, -1.443,
       -1.394, -1.386, -1.269, -1.195, -1.174, -0.854, -0.620,
       -0.576, -0.548, -0.247, -0.195, -0.056, -0.013,  0.006,
        0.033,  0.037,  0.046,  0.084,  0.221,  0.245,  0.296)
D <- c(D,rep(.2960001,40-28)) # 28 values, but Hosking mentions
                              # 40 values in total
z <-  pwmRC(D,threshold=.2960001)
str(z)
# Hosking reports B-type L-moments for this sample are
# lamB1 = -0.516 and lamB2 = 0.523
btypelmoms <- pwm2lmom(z$Bbetas)
# My version of R reports lamB1 = -0.5162 and lamB2 = 0.5218
str(btypelmoms)
rg.pars <- parrevgum(btypelmoms,z$zeta)
str(rg.pars)
# Hosking reports xi=0.1636 and alpha=0.9252 for the sample
# My version of R reports xi = 0.1635 and alpha = 0.9254
# Now one can continue one with a plotting example.
## Not run: 
F  <- nonexceeds()
PP <- pp(D) # plotting positions of the data
D  <- sort(D)
plot(D,PP)
lines(D,cdfrevgum(D,rg.pars))
# Now finally do the PDF
F <- seq(0.01,0.99,by=.01)
x <- quarevgum(F,rg.pars)
plot(x,pdfrevgum(x,rg.pars),type='l')

## End(Not run)

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